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Conchoid of Nicomedes (conchoid of a straight line) for various values of the set distance. The black line is the reference curve (C in the definition), and the red dot is the fixed point (P).
A conchoid of circle
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From Latin concha (mussel) (from Ancient Greek κόνχη (kónkhē)) +‎ -oid, referring to the curved outline of a mussel shell.


conchoid (plural conchoids)

  1. (mathematics, geometry) Any of a family of curves defined as the locus of points p, such that each p is on a line that passes through a given fixed point P and intersects a given curve, C, and the distance from p to the point of intersection with C is a specified constant (note that for nontrivial cases two such points p satisfy the criteria, and the resultant curve has two parts).
    The conchoid of a circle with respect to a point on the circle is a cardioid if the fixed distance is equal to the diameter of the circle.
    The Conchoid of Nicomedes is the conchoid of a straight line with respect to a point not on the line.
    • 1815, Charles Hutton, Pappus, entry in A Philosophical and Mathematical Dictionary, Volume 2, page 147,
      He next treats of the properties of the Conchoid, which Nicomedes invented for doubling the cube; applying it to the solution of certain problems concerning Inclinations, with the finding of two mean proportionals, and cubes in any proportion whatever.
    • 1982, J. Lee Kavanau, Curves and Symmetry, Volume 1, page 3,
      The classical conchoid construction is a non-orthogonal polar-curvilinear construction in which equal distances along a line are marked off from its point of intersection with a curve for various positions of the line as it rotates about a point.
    • 2007, James Stewart, Single Variable Calculus, Volume 2, page 662,
      These curves are called conchoids of Nicomedes after the ancient Greek scholar Nicomedes. He called them conchoids because the shape of their outer branches resembles that of a conch shell or mussel shell.
    • 2009, Niccolò Guicciardini, Isaac Newton on Mathematical Certainty and Method, page 68,
      One of the best choices is the conchoid, according to Newton the simplest curve after the circle.
  2. (geology) A conchoidal fracture in rock.
    • 1948, Tennessee Valley Authority, The Hiwassee Valley Projects, Technical Report, Issue 5, Volume 2, page 359,
      Conchoids of sound rock, from a few feet to 20 or more feet in diameter, entirely surrounded by comparatively thin layers of weathered material, were frequently encountered, sometimes in adjacent series.

Usage notes[edit]

The fixed point (P) of the construction may be referred to as the focus of the conchoid; it may also be defined as the origin (of a Cartesian coordinate system) or the pole (if polar coordinates are used), and potentially referred to accordingly. The curve C is an example of a directrix.

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